Weil restriction

In mathematics, specifically the theory of algebraic groups, Weil restriction is a functor allowing one to pass from an algebraic group G over a field L to another one, RG, over a subfield K. The idea is that the group of points G(L) of G over L should be deemed RG(K).

For example taking L = C to be the complex number field, and K = R the real number field, we can apply Weil restriction to the multiplicative group

GL₁

to get

R_C/RGL₁,

which is a two-dimensional algebraic group. It consists of 2×2 matrices of the shape that is given by the action of a+bi on the basis {1,i} of C over R:

This group is an algebraic torus, and is applied in Hodge theory, where its linear representations are Hodge structures.

Note that the construction is of an algebraic variety, not just a set of points: a group object, not simply a group. To say this more formally, we should identify RG as a right adjoint. There is an extension of scalars

E_L/K

functor to which it is adjoint. For any K-algebra A we have

E_L/K(A)

the tensor product of A with L over K (as K-vector spaces), which is made into an L-algebra using the existing ring product in A and in L. Then it is almost true to say that R_L/K is the right adjoint to E_L/K.

To be completely accurate, we should do this: an algebraic group H over K is such that for a commutative K-algebra B, H(B) is

Hom (Spec(B), H)

in a suitable category (of schemess over Spec(K)). Another way of putting it is that Spec makes the category of commutative K-algebras into its opposite. Therefore the actual adjunction relation is of the type

Hom(ESpec(B), G) = Hom(Spec(B), RG)

where on the left side we are in the opposite of the category of commutative L-algebras, on the right side in the opposite of the category of commutative K-algebras, and E becomes the fiber product over Spec(K) with Spec(L). This is a complete definition in the case that G is an affine algebraic group.

The case where G is an abelian variety is also of importance, though. It is one non-trivial way to construct higher-dimensional abelian varieties from elliptic curves, for example. Weil restriction multiplies dimension by the degree [L:K], as one can compute with the tangent space (in characteristic 0).

The Weil restriction is essential for the classification of algebraic groups over fields that are not algebraically closed.

NodeWorks boosts web surfing!

Page Returned in 0.164 seconds - HTML Compressed 69.9%

This article is from Wikipedia. All text is available
under the terms of the GNU Free Documentation License.